AN ALTERNATE APPROACH TO THE LIE BRACKET ON
HOCHSCHILD COHOMOLOGY
CRIS NEGRON AND SARAH WITHERSPOON
Abstract. We define Gerstenhaber’s graded Lie bracket directly on complexes
other than the bar complex, under some conditions, resulting in a practical technique for explicit computations. The Koszul complex of a Koszul algebra in particular satisfies our conditions. As examples we recover the Schouten-Nijenhuis bracket
for a polynomial ring and the Gerstenhaber bracket for a group algebra of a cyclic
group of prime order.
1. Introduction
Hochschild cohomology incorporates useful information about an algebra, the study
of which was begun by Hochschild [13] and Gerstenhaber [8]. In low degrees one
finds the center of the algebra, derivations, infinitesimal deformations, and obstructions in algebraic deformation theory. Vanishing in high degrees is equivalent to
smoothness in commutative settings [1, 2]. Noncommutative algebras can behave
quite differently [3], yet analogous notions have also been explored in noncommutative settings [14]. Hochschild cohomology is used in support variety theory, a tool for
studying representations of some types of finite dimensional algebras [21].
In spite of its many uses, some of the structure of Hochschild cohomology remains
elusive. It is a Gerstenhaber algebra, that is, it has both a cup product and a graded
Lie bracket, and the bracket induces graded derivations with respect to the product.
Products are defined in any number of equivalent ways: as Yoneda composition of
n-extensions of bimodules, as composition of maps in arbitrary projective bimodule
resolutions, or as application of a diagonal map to the tensor product over the algebra
of two copies of an arbitrary bimodule resolution. This freedom of choice makes the
product quite tractable for many algebras. Brackets have not been so amenable to
study on resolutions other than the bar resolution where they were historically defined,
and thus they are more difficult to compute and to use. Typically one computes
cohomology with a resolution other than the bar resolution, and then translates the
bracket from the bar resolution using explicit comparison maps. These maps are
nearly always very cumbersome, and beg for a better approach.
Date: 10 October 2015.
2010 Mathematics Subject Classification. 16E40, 18G10, 16S37.
Key words and phrases. Hochschild cohomology, Gerstenhaber bracket, Koszul algebra.
The first author was supported by the NSF Graduate Research Fellowship under grant DGE1256082. The second author was partially supported by NSF grant DMS-1101399.
1
2
CRIS NEGRON AND SARAH WITHERSPOON
The question about realizing the Gerstenhaber bracket on other resolutions was
raised by Gerstenhaber and Schack [9]. An elegant such realization was given by
Schwede [20], based on Retakh’s description of categories of extensions [18]. Hermann [12] generalized Schwede’s construction of brackets as loops in an extension
category to other suitable exact monoidal categories. Yet it seems difficult to translate these beautiful constructions into practical techniques for explicit computations
of brackets as may be required, for example, to answer some questions in algebraic
deformation theory. Our paper takes a different route to computational techniques.
We begin with the observation that there is more than one way to define the
graded Lie bracket on the bar resolution B of an algebra: We show in Section 2 that
a particular class of chain maps, of graded degree 1, from the tensor product of two
copies of B to B, gives rise to many brackets at the chain level. These all induce the
Gerstenhaber bracket on cohomology. We mimic this construction in Section 3 for
other resolutions satisfying some hypotheses. We define brackets and prove that these
brackets also induce Gerstenhaber brackets on cohomology. One useful condition in
particular is when the resolution embeds into the bar resolution in such a way that the
diagonal maps commute with the embedding, and the strongest results follow from
this condition (Subsection 3.2). Koszul resolutions of Koszul algebras in particular
satisfy this hypothesis.
We illustrate our techniques by recovering the Schouten-Nijenhuis bracket on polynomial rings in Section 4. We also give some results under weaker conditions (Subsection 3.4) that still may be useful but have the disadvantage of requiring a more
detailed comparison with the bar resolution. In Section 5 we show that these techniques may be used to recover Gerstenhaber brackets for a group algebra of a cyclic
group of prime order p over a field of characteristic p. (Expressions for such brackets
were originally given in the work of Sanchez-Flores [19].) These two well known classes
of examples, in Sections 4 and 5, serve merely to illustrate our techniques here. A
new class of examples is given in [10]: Brackets are computed there for the quantum
complete intesections Λq := khx, yi/(x2 , y 2 , xy + qyx) for various (nonzero) values of
a parameter q in a field k. The algebra structure of Hochschild cohomology of Λq had
been computed by Buchweitz, Green, Madsen, and Solberg [3]. Grimley, Nguyen, and
the second author [10] used the techniques of the current paper to compute Gerstenhaber brackets directly on the Koszul resolution of Λq . They did not need to know
explicit formulas for chain maps between the bar and Koszul resolutions, as these were
not used; it suffices to know existence of such maps satisfying some conditions. Also
in [10] is a general result about the Gerstenhaber algebra structure of the Hochschild
cohomology of a twisted tensor product of algebras. Its proof uses techniques from
the current paper, showing that these techniques can be useful as well for algebras
that are not Koszul.
2. Alternate brackets on the Hochschild complex
Let k be a field of arbitrary characteristic and let A be a k-algebra. Let us recall
the definitions of the bar resolution B of A and the Hochschild cochain complex. We
write ⊗ to mean ⊗k .
LIE BRACKET ON HOCHSCHILD COHOMOLOGY
3
Let T A = T (A) denote the graded tensor coalgebra, that is, T A = ⊕r≥0 (T A)r
where (T A)r = A⊗r and the coproduct ∆ : T A → T A ⊗ T A is the k-linear map
defined by
∆(a1 ⊗ · · · ⊗ ar ) =
r
X
(a1 ⊗ · · · ⊗ ai ) ⊗ (ai+1 ⊗ · · · ⊗ ar )
i=0
for a1 , . . . , ar ∈ A. As a graded A-bimodule, we have B = A ⊗ T A ⊗ A, with
Br = A⊗(r+2) for each r ≥ 0. We may use the notation a ⊗ x ⊗ a0 to denote monomials
in B, where a, a0 ∈ A, and x ∈ T A. The differential on B is
X
(−1)i a0 ⊗ . . . ⊗ ai ai+1 ⊗ . . . ⊗ ar+1 .
(2.0.1)
a0 ⊗ . . . ⊗ ar+1 7→
0≤i≤r
Note that the comultiplication on T A induces a quasi-isomorphism
a0 ⊗ . . . ⊗ ar+1
X ∆ : B → B ⊗A B
7→
(a0 ⊗ . . . ⊗ ai ⊗ 1) ⊗ (1 ⊗ ai+1 ⊗ . . . ⊗ ar+1 ).
(2.0.2)
i
The map ∆ is coassociative by construction, that is, (∆ ⊗ id)∆ = (id ⊗ ∆)∆ as chain
maps from B to B ⊗A B ⊗A B. On monomials, we can write this map symbolically as
X
a ⊗ x ⊗ a0 7→
(a ⊗ x1 ⊗ 1) ⊗ (1 ⊗ x2 ⊗ a0 )
where the sum runs over all possible ways to factor the monomial x. Let
C(A) := HomAe (B, A),
where Ae = A ⊗ Aop . The cup product may be defined at the cochain level via the
diagonal map ∆: If f ∈ HomAe (Br , A), g ∈ HomAe (Bs , A), then
(f ^ g)(a0 ⊗ . . . ⊗ ar+s+1 ) = f (a0 ⊗ . . . ⊗ ar ⊗ 1)g(1 ⊗ ar+1 ⊗ . . . ⊗ ar+s+1 ).
We use the notation |f | = r for the degree of f in this case. The Gerstenhaber bracket
is defined as follows, where we employ for the moment the canonical identifications
HomAe (Bn , A) = Homk (A⊗n , A) via freeness of Bn .
Definition 2.0.3 (Standard Gerstenhaber Bracket [8]). Let ◦ denote the operation
C(A) ⊗ C(A) → C(A) given on homogeneous elements f and g by
f ◦ g(a1 ⊗ . . . ⊗ an )
|f |
X
=
(−1)(|g|−1)(j−1) f (a1 ⊗ . . . ⊗ aj−1 ⊗ g(aj ⊗ . . . ⊗ aj+|g|−1 ) ⊗ aj+|g| ⊗ . . . ⊗ an ),
j=1
and define the bracket [ , ] by
[f, g] = f ◦ g − (−1)(|f |−1)(|g|−1) g ◦ f.
The cup product and bracket induce operations on Hochschild cohomology that
enjoy many useful properties, for example,
¯
[f¯ ^ ḡ, h̄] = [f¯, h̄] ^ ḡ + (−1)|f |(|h̄|−1) f¯ ^ [ḡ, h̄],
(2.0.4)
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CRIS NEGRON AND SARAH WITHERSPOON
where f, g, h are homogeneous cocycles and f¯, ḡ, h̄ are their images in Hochschild
cohomology. See [8] for this and other properties.
Lemma 2.0.5. As a graded A-bimodule, B ⊗A B ∼
= A ⊗ T A ⊗ A ⊗ T A ⊗ A. Under
this identification, the differential is given by
(a0 ⊗ . .P
. ⊗ aj−1 ) ⊗ (aj ) ⊗ (aj+1 ⊗ . . . ⊗ an+1 )
i
i<j−1 (−1) (a0 ⊗ . . . ⊗ ai ai+1 . . . ) ⊗ (aj ) ⊗ (. . . ⊗ an+1 )
j−1
+(−1) (a0 ⊗ . . . ) ⊗ (aj−1 aj ) ⊗ (. . . ⊗ an+1 )
7→
j−1 (a ⊗ . . . ) ⊗ (a a
+(−1)
0
j j+1 ) ⊗ (. . . ⊗ an+1 )
P
+ k>j (−1)k−1 (a0 ⊗ . . . ) ⊗ (aj ) ⊗ (. . . ⊗ ak ak+1 . . . ⊗ an+1 ).
Proof. The first portion of the statement is clear. The second is an easy check from
the fact that the differential on the tensor complex X ⊗A Y , of any two A-bimodule
complexes X and Y , is given by d(x ⊗ y) = d(x) ⊗ y + (−1)|x| x ⊗ d(y).
We will deconstruct the bracket operation, realize it as a composition of several
maps, and make some changes in the apparent choices involved. We will observe that
these choices do not matter at the level of cohomology, giving us some freedom in
the definition. It is this freedom that will allow us, in the next section, to define the
bracket independently on other cochain complexes satisfying certain conditions.
We first define a chain map FB : B ⊗A B → B. By the isomorphism of Lemma 2.0.5,
elements in the tensor product B ⊗A B may be identified with sums of elements of
the form
a ⊗ x ⊗ a0 ⊗ y ⊗ a00 ,
with x ∈ A⊗i , y ∈ A⊗j and a, a0 , a00 ∈ A. We define FB : B ⊗A B → B on such
monomials as follows: If i > 0 and j > 0, then
FB (a ⊗ x ⊗ a0 ⊗ y ⊗ a00 ) = 0,
FB (a ⊗ a0 ⊗ y ⊗ a00 ) = aa0 ⊗ y ⊗ a00 ,
FB (a ⊗ x ⊗ a0 ⊗ a00 ) = −a ⊗ x ⊗ a0 a00 .
(2.0.6)
In degree 0,
FB (a ⊗ a0 ⊗ a00 ) = aa0 ⊗ a00 − a ⊗ a0 a00 .
As one can see from the definition, FB is 0 on most of the tensor complex B ⊗A B,
and is simply given by the actions of A on B for the extremal terms B 0 ⊗A B j and
B i ⊗A B 0 . One may check directly that FB is a chain map. Alternatively, this follows
from Proposition 2.0.8 or the general construction given in Section 3.2.
In the remainder of this article, we use the isomorphism of Lemma 2.0.5, without
comment, to identify B ⊗A B with A ⊗ T A ⊗ A ⊗ T A ⊗ A.
Notation 2.0.7.
(1) Let ∆(2) denote the map
(∆ ⊗ id)∆ = (id ⊗ ∆)∆ : B → B ⊗A B ⊗A B.
(2) Let G : B ⊗A B → B denote the map given on monomials by
G (a0 ⊗ . . . ⊗ aj−1 ) ⊗ (aj ) ⊗ (aj+1 ⊗ . . . ⊗ an+1 )
= (−1)j−1 a0 ⊗ . . . ⊗ aj−1 ⊗ aj ⊗ aj+1 ⊗ . . . ⊗ an+1 .
LIE BRACKET ON HOCHSCHILD COHOMOLOGY
5
Notice that the circle operation f ◦ g of Definition 2.0.3, on Hochschild cochains f
and g is precisely the composition
∆(2)
id ⊗ g⊗ id
G
f
B A
A B
B −−→ B ⊗A B ⊗A B −−−−−−−−→
B ⊗A B −→ B −→ A.
To be clear, in the definition of the map idB ⊗A g ⊗A idB one includes “Koszul signs”
so that on elements the map is given by
(a ⊗ x ⊗ a0 ⊗ y ⊗ a00 ⊗ z ⊗ a000 ) 7→ (−1)|x||g| (a ⊗ x) ⊗ g(a0 ⊗ y ⊗ a00 ) ⊗ (z ⊗ a000 ).
This observation inspires our alternate definition of brackets below (Definition 2.1.1).
First we record a crucial property of the map G.
Proposition 2.0.8. Let d be the differential on the complex HomAe (B ⊗A B, B).
The map G ∈ HomAe (B ⊗A B, B) is a contracting homotopy for FB , that is, d(G) :=
dB G + GdB⊗A B = FB .
Proof. Take a monomial
(a0 ⊗ . . . ⊗ aj−1 ) ⊗ (aj ) ⊗ (aj+1 ⊗ . . . ⊗ an+1 ) ∈ Bj−1 ⊗A Bn−j
with j − 1, n − j > 0. Applying the formulas given in Lemma 2.0.5 and Notation 2.0.7(2), the function GdB⊗A B sends (a0 ⊗ . . . ⊗ aj−1 ) ⊗ (aj ) ⊗ (aj+1 ⊗ . . . ⊗ an+1 )
to the element
P
(i+j−2) (a ⊗ . . . ⊗ a a
0
i i+1 ⊗ . . . ⊗ an+1 )
i<j−1 (−1)
(j−1+j−2)
+(−1)
(a0 ⊗ . . . ⊗ aj−1 aj ⊗ . . . ⊗ an+1 )
(2.0.9)
+(−1)(j−1+j−1) (a0 ⊗ . . . ⊗ aj aj+1 ⊗ . . . ⊗ an+1 )
P
+ j<k (−1)(k−1+j−1) (a0 ⊗ . . . ⊗ ak ak+1 ⊗ . . . ⊗ an+1 )
in B. Now dB G will send that same element in B ⊗A B to
P
(j−1+i) (a ⊗ . . . ⊗ a a
0
i i+1 ⊗ . . . ⊗ an+1 )
i<j−1 (−1)
(j−1+j−1)
+(−1)
(a0 ⊗ . . . ⊗ aj−1 aj ⊗ . . . ⊗ an+1 )
+(−1)(j−1+j) (a0 ⊗ . . . ⊗ aj aj+1 ⊗ . . . ⊗ an+1 )
P
+ j<k (−1)(j−1+k) (a0 ⊗ . . . ⊗ ak ak+1 ⊗ . . . ⊗ an+1 ).
(2.0.10)
Comparing the exponents of −1, we see that GdB⊗A B = −dB G so d(G) = 0 on
B>0 ⊗A B>0 .
Now consider an element (a0 ) ⊗ (a1 ) ⊗ (a2 ⊗ . . . ⊗ an+1 ) ∈ B0 ⊗A Bn−1 , for which
n − 1 > 0. Applying GdB⊗A B to this element yields (2.0.9) where j = 1, minus the
first two summands, and applying dB G yields (2.0.10) where j = 1, minus the first
summand. So applying d(G) to this element yields
a0 a1 ⊗ . . . ⊗ an+1 ∈ B.
Similarly, for the elements (a0 ⊗ . . . ⊗ an−1 ) ⊗ (an ) ⊗ (an+1 ), applying d(G) yields
(−1)n−1+n a0 ⊗ . . . ⊗ an an+1 = −a0 ⊗ . . . ⊗ an an+1 .
In degree 0 we have
d(G)((a0 ) ⊗ (a1 ) ⊗ (a2 )) = a0 a1 ⊗ a2 − a0 ⊗ a1 a2 .
6
CRIS NEGRON AND SARAH WITHERSPOON
Comparing these values in the different cases to our definition of FB above, we see
that d(G) = FB .
2.1. Alternate definition of bracket on the Hochschild complex. We call a
map φ : B ⊗A B → B for which d(φ) := dB φ + φdB⊗A B = FB a contracting homotopy
for FB .
Definition 2.1.1 (φ-circle operation, φ-bracket). Let φ ∈ HomAe (B ⊗A B, B) be any
contracting homotopy for FB . The φ-circle operation f ◦φ g on Hochschild cochains
is defined as the composite
f ◦φ g := f φ(idB ⊗A g ⊗A idB )∆(2) .
The φ-bracket is then defined as the graded commutator
[f, g]φ := f ◦φ g − (−1)(|f |−1)(|g|−1) g ◦φ f.
Note that the G-circle operation ◦G is the standard circle operation and the Gbracket [ , ]G is the standard Gerstenhaber bracket.
Lemma 2.1.2. Let φ be a contracting homotopy for FB . The difference (φ − G) :
B ⊗A B → B is a boundary in the Hom complex.
Proof. The difference is a cycle, since d(φ) = d(G) = FB . Recall that the following
map is a quasi-isomorphism, where proj∗ is induced by the canonical projection of B
onto A (considered as a complex in degree 0, with 0 in all other degrees):
proj
∗
HomAe (B ⊗A B, B) −−−→
HomAe (B ⊗A B, A).
Note that B ⊗A B is also a bimodule resolution of A, by the Künneth formula, and
so the homology of the right hand side is ExtAe (A, A). Since ExtAe (A, A) is 0 in
negative degrees, we have H−1 (HomAe (B ⊗A B, B)) = 0. So any cycle in degree −1
is a boundary. Consequently, the difference φ − G is a boundary.
Proposition 2.1.3. Let f and g be cocycles in the cochain complex C(A) = HomAe (B, A).
Let φ be a contracting homotopy for FB . Then the difference
f ◦φ g − f ◦ g
is a boundary, as is the difference
[f, g]φ − [f, g].
Proof. Take g̃ to be the function idB ⊗A g ⊗A idB in HomAe (B ⊗A B ⊗A B, B ⊗A B).
By this notation g̃, in degree n, we mean the sum of all maps idi ⊗A g ⊗A idj on
(B ⊗A B ⊗A B)n , where idi is the identity map on Bi and i + j = n − |g|. Note that
the map g̃ is still a cocycle since g is a cocycle. Then
f ◦φ g = f φg̃∆(2) , while f ◦ g = f Gg̃∆(2) .
The difference is given by
f ◦φ g − f ◦ g = f φg̃∆(2) − f Gg̃∆(2)
= f (φ − G)g̃∆(2) .
LIE BRACKET ON HOCHSCHILD COHOMOLOGY
7
By Lemma 2.1.2, there exists some map ψ with d(ψ) = φ − G. Then, since f and g̃
are cocycles,
(−1)|f | d(f ψg̃∆(2) ) = f d(ψ)g̃∆(2)
= f (φ − G)g̃∆(2)
= f ◦φ g − f ◦ g,
whence f ◦φ g − f ◦ g is seen to be a boundary, as claimed. The second statement
follows from the first.
Corollary 2.1.4.
(1) For any two cocycles f and g, the φ-bracket [f, g]φ is yet
another cocycle.
(2) If f or g is a boundary, then so is [f, g]φ .
(3) On cocycles, the φ-bracket is graded anti-commutative up to a boundary and
also satisfies the Jacobi identity up to a boundary.
Proof. All of these statements follow from the previous proposition and the fact that
these conditions are satisfied by the Gerstenhaber bracket.
Corollary 2.1.5. For any contracting homotopy φ for FB , the φ-bracket [ , ]φ induces
a graded Lie bracket on the shifted cohomology
[ , ]φ : HH(A)[1] ⊗ HH(A)[1] → HH(A)[1].
This bracket agrees with the standard Gerstenhaber bracket on cohomology.
3. Brackets on other cochain complexes
In this section we define brackets at the cochain level on complexes other than
the Hochschild complex. We show that under some conditions, these brackets induce
precisely the Gerstenhaber bracket on cohomology. Koszul algebras over k will satisfy
these conditions.
Let K → A be a projective A-bimodule resolution of A. For most of this section
we will want K to satisfy some hypotheses which we outline next.
3.1. Hypotheses on the bimodule resolution K → A. We assume that the Abimodule resolution K → A satisfies the following conditions:
(a) K admits an embedding ι : K → B of complexes of A-bimodules for which
the following diagram commutes
ι
K
8B
&/
A.
(b) The embedding ι admits a section π : B → K, i.e. an Ae -chain map π with
πι = idK .
(c) The diagonal map ∆B : B → B ⊗A B preserves K, and hence induces a
diagonal quasi-isomorphism ∆K : K → K ⊗A K. Equivalently, K comes
equipped with a diagonal quasi-isomorphism ∆K : K → K ⊗A K satisfying
∆B ι = (ι ⊗A ι)∆K .
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CRIS NEGRON AND SARAH WITHERSPOON
Practically speaking, the easiest way for condition (b) to be satisfied is for K to
be free on some graded base space W ⊂ K with W mapping to T A ⊂ B under ι.
Indeed, one can verify that condition (b) holds if and only if the cokernel of each map
ιl : Kl → Bl is projective over Ae . So we could, alternately, require that K satisfy
the slightly stronger condition
(b0 ) K is free on a graded base W ⊂ K with ι(W ) ⊂ T A in B.
Conditions (a) and (b) can be seen as relatively mild restrictions. In contrast,
condition (c) holds a great deal of significance. Indeed, it can be shown that if
the minimal free bimodule resolution of a connected graded algebra can be made to
satisfy (c), then the algebra is Koszul. This does not mean, however, that non-Koszul
algebras have no resolutions satisfying the above conditions, or that the minimal
resolution can not be used in some way to compute the Lie bracket. We will see
in Section 5 that we can still use the minimal resolution for (the group algebras
of) the cyclic p-group in characteristic p to compute the Lie bracket on Hochschild
cohomology.
As the above discussion suggests, the Koszul complex of a Koszul algebra does
satisfy our conditions (a)–(c). See, e.g., [4] or [16] for a discussion of diagonal maps
in the case of a Koszul algebra. Verification of the other conditions is more straightforward. We will not need the definition of a Koszul algebra however, as we work in
the general setting of a complex satisfying conditions (a)–(c). In the next section we
give explicitly the example of a polynomial ring, which is a Koszul algebra. One can
also show that the Koszul resolution of a PBW deformation of a Koszul algebra fits
into our framework ([7, Lemma 4.1], [16, Lemma 6.2], [17]). In this case, the diagonal map on K will be induced by the natural comultiplication on the base W ⊂ K.
Localizations of such algebras will also fit into our scheme.
Remark 3.1.1. One actually has to replace B with the reduced bar resolution to get
(a)–(c) to hold in the case of a non-augmented PBW deformation of a Koszul algebra.
This is, however, a straightforward process.
3.2. φ-brackets on HomAe (K, A). For this subsection, let us fix a resolution K → A
satisfying the hypotheses 3.1(a)–(c). Let µ denote the given quasi-isomorphism µ :
K → A. Then we have the two chain maps µ ⊗A idK : K ⊗A K → A ⊗A K ∼
= K and
idK ⊗A µ : K ⊗A K → K ⊗A A ∼
= A. We define the chain map FK as the difference
of these two maps,
FK := (µ ⊗A idK − idK ⊗A µ) : K ⊗A K → K.
(The natural isomorphisms A ⊗A K ∼
= K and K ⊗A A ∼
= K implicit in the above
definition have been omitted from the notation.) In the case that K satisfies (b0 ),
so that K = A ⊗ W ⊗ A and the elements in K are given by sums of monomials
a ⊗ x ⊗ a0 , we get the elementwise definition of FK analogous to the one given in
(2.0.6). In particular, when K = B the two definitions of FB agree.
Lemma 3.2.1. The map FK : K ⊗A K → K is a boundary in HomAe (K ⊗A K, K).
Proof. It suffices to check that FK maps to 0 under the quasi-isomorphism
µ∗
HomAe (K ⊗A K, K) −−→ HomAe (K ⊗A K, A).
LIE BRACKET ON HOCHSCHILD COHOMOLOGY
9
Since FK is the difference of the two maps µ ⊗A idK and idK ⊗A µ, composed with
the isomorphisms A ⊗A K ∼
= K and K ⊗A A ∼
= K respectively, the image of FK is 0
if and only if the images of these two maps agree.
∼
∼
For any complex M of A-bimodules, let ϕM : A⊗A M → M and ϕ0M : M ⊗A A → M
denote the standard isomorphisms. Since these isomorphisms are natural we will have
a commutative diagram
K ⊗A K
µ⊗A idK
/ A ⊗A K
idA ⊗A µ
/ A ⊗A A
ϕK
K
µ
ϕA
/ A.
Whence
µ∗ ϕK (µ ⊗A idK )
= µϕK (µ ⊗A idK )
= ϕA (idA ⊗A µ)(µ ⊗A idK )
= ϕA (µ ⊗A µ).
Similarly, we see µ∗ ϕ0K (idA ⊗A A) = ϕ0A (µ⊗A µ). Since there is an equality ϕA = ϕ0A ,
this gives the desired equality
µ∗ ϕK (µ ⊗A idK ) = µ∗ ϕ0K (idA ⊗A A) ,
and we conclude µ∗ (FK ) = 0.
Recall that a contracting homotopy for FK is a map φ : K ⊗A K → K for which
d(φ) := dK φ + φdK⊗A K = FK . The lemma allows us to make the following definition.
Definition 3.2.2 (General φ-circle operation, φ-bracket). Let φ be a contracting
(2)
homotopy for FK , and let ∆K be a chain map from K to K ⊗A K ⊗A K. (Under
(2)
hypothesis 3.1(c), we take ∆K := (∆K ⊗ idK )∆K = (idK ⊗ ∆K )∆K .) The φ-circle
product f ◦φ g is the composition
(2)
f ◦φ g := f φ(idK ⊗A g ⊗A idK )∆K .
The φ-bracket is the graded commutator
[f, g]φ := f ◦φ g − (−1)(|f |−1)(|g|−1) g ◦φ f.
Suppose that our resolution K → A satisfies the freeness property (b0 ). For example, we could take K to be the Koszul resolution of a PBW deformation of a Koszul
algebra. (See, e.g., [7, 17].) We can then express the φ-circle operation and bracket
on elements in the generating set x ∈ W ⊂ K as
X
(f ◦φ g)(x) =
(−1)|g||x1 | f φ(x1 ⊗ g(x2 ) ⊗ x3 )
and
[f, g]φ (x) =
X
(−1)|g||x1 | f φ(x1 ⊗ g(x2 ) ⊗ x3 )
X
−(−1)(|g|−1)(|f |−1)
(−1)|f ||x1 | g φ(x1 ⊗ f (x2 ) ⊗ x3 ) .
10
CRIS NEGRON AND SARAH WITHERSPOON
P
Here the sum x1 ⊗ x2 ⊗ x3 denotes the element ∆(2) (x), which lies in W ⊗ W ⊗ W ⊂
K ⊗A 3 by hypothesis. In the case that K = B and φ = G, the map φ simply inserts
the apparent missing factor (−1)|x1 | in the above expressions.
We will see in Theorem 3.2.5 that the φ-bracket operation preserves cocycles and
coboundaries, and that the induced operation on cohomology is precisely the Gerstenhaber bracket. The following lemma will be of significance in a moment.
Lemma 3.2.3. Let us take GK := πG(ι⊗A ι) : K ⊗A K → K, where G is the standard
contracting homotopy for FB given in Notation 2.0.7(2). Then
(1) FK = πFB (ι ⊗A ι).
(2) d(GK ) = FK .
Proof. Statement (1) follows directly from the definitions of FK and FB given above,
the commutative diagram
8B
ι
&/
K
A
of hypothesis 3.1(a), and the fact that πι = idK . Statement (2) follows from (1) since
we have
d(GK ) = d(πG(ι ⊗A ι)) = πd(G)(ι ⊗A ι) = πFB (ι ⊗A ι) = FK .
Since π : B → K and ι : K → B are quasi-isomorphisms, they induce quasiisomorphisms on the Hom complexes
π ∗ : HomAe (K, A) → HomAe (B, A) and ι∗ : HomAe (B, A) → HomAe (K, A).
The latter map is simply restriction to K.
Proposition 3.2.4. Assume hypotheses 3.1(a)–(c). Given f and g in HomAe (K, A)
we have an equality of functions
f ◦GK g = ι∗ (π ∗ f ◦ π ∗ g)
and subsequent equality
[f, g]GK = ι∗ [π ∗ f, π ∗ g].
Proof. Let us simply expand the functions.
(2)
(f π ◦ gπ)ι = f πG(idB ⊗A gπ ⊗A idB )∆B ι
(2)
= f πG(idB ⊗A gπ ⊗A idB )ι⊗3 ∆K
(2)
= f πG(ι ⊗A g(πι) ⊗A ι)∆K
(2)
= f (πG(ι ⊗A ι))(idK ⊗A g ⊗A idK )∆K (since πι = idK )
(2)
= f GK (idK ⊗A g ⊗A idK )∆K
= f ◦GK g.
Equality of brackets follows from the definition of the bracket as the graded ◦commutator.
LIE BRACKET ON HOCHSCHILD COHOMOLOGY
11
Let φ be any contracting homotopy for FK . By the same proof as the one given
for Proposition 2.1.3, the differences
f ◦φ g − f ◦GK g and [f, g]φ − [f, g]GK
will be boundaries whenever f and g are cocycles in HomAe (K, A).
Theorem 3.2.5. Suppose K is a bimodule resolution of A satisfying hypotheses 3.1(a)–
(c), and let φ be any contracting homotopy for FK . Let f and g be cocycles in
HomAe (K, A).
(1) The bracket [f, g]φ is a cocycle.
(2) If f or g is a boundary, then [f, g]φ is a boundary.
(3) The induced bracket [ , ]φ : HH(A)[1] ⊗ HH(A)[1] → HH(A)[1] on cohomology
agrees with the Gerstenhaber bracket.
Proof. By the discussion preceding the statement of the theorem, we may assume
without loss of generality that φ = GK = πG(ι ⊗A ι). Now (1) and (2) follow directly
from Proposition 3.2.4 and the fact that π ∗ and ι∗ are quasi-isomorphisms. Since
idH(Hom(K,A)) = (πι)∗ = ι∗ π ∗ , we see that the induced isomorphisms on homology are
mutually inverse. So we have
∼
=
(ι∗ )−1 = π ∗ : H(HomAe (K, A)) −→ H(HomAe (B, A)).
This isomorphism is one of graded Lie algebras since, according to Proposition 3.2.4,
we will have an equality
π ∗ ([f, g]GK ) = π ∗ (ι∗ [π ∗ f, π ∗ g]) = (π ∗ ι∗ )[π ∗ f, π ∗ g] = [π ∗ f, π ∗ g]
on cohomology. Finally, the homologies H(HomAe (K, A)) and H(HomAe (B, A)) are
precisely the Hochschild cohomology HH(A).
3.3. Formula for φ. In general, it may be difficult to find a map φ satisfying d(φ) :=
dK φ + φdK⊗A K = FK . Let us give one method for constructing such a homotopy that
is related to constructions of chain maps via contracting homotopies (for example, as
described in Mac Lane [15]). Consider the extended complex K → A → 0, by which
we mean the complex · · · → K1 → K0 → A → 0 with A in degree −1.
The following lemma is general, that is, it does not require hypotheses 3.1, only
that K be a free A-bimodule resolution of A, as well as the further hypotheses stated
in the lemma.
Lemma 3.3.1. Suppose K is free on a graded subspace W ⊂ K. Let h be any k-linear
contracting homotopy for the identity map on the extended complex K → A → 0. Take
φ−1 = 0. Define φ, in each degree i ≥ 0, as the Ae -linear map φi : (K ⊗A K)i → Ki+1
given inductively by the formula
φi |W ⊗A⊗W := hi ((FK )i − φi−1 (dK⊗A K )i )|W ⊗A⊗W .
Then d(φ) = FK .
Proof. To simplify notation, take F = FK and d = dK , or dK⊗A K when appropriate.
Let us consider F and φ as maps to the extended complex. Note that, since F0 has
image in the space of degree 0 cycles Z0 (K → A → 0), this new version of F will still
12
CRIS NEGRON AND SARAH WITHERSPOON
be a chain map. Take φj = 0 for all negative j. Then for all negative j the equality
dj+1 φj + φj−1 dj = Fj holds, since both sides are just 0. Now suppose, for a given i,
that for all j < i the formula dj+1 φj + φj−1 dj = Fj holds. Then after restricting to
the generating subspace W ⊗ A ⊗ W ⊂ K ⊗A K,
(di+1 φi + φi−1 di ) = di+1 hi (Fi − φi−1 di ) + φi−1 di
= (Fi − φi−1 di ) − hi−1 di (Fi − φi−1 di ) + φi−1 di
= Fi − φi−1 di − hi−1 di Fi + hi−1 di φi−1 di + φi−1 di
= Fi − hi−1 di Fi + hi−1 di φi−1 di
= Fi − hi−1 di Fi + hi−1 Fi−1 di − hi−1 φi−2 di−1 di
= Fi − hi−1 di Fi + hi−1 Fi−1 di
= Fi ,
whence d(φ) = F .
We will use the formula of Lemma 3.3.1 in Sections 4 and 5.
3.4. φ-brackets under weaker conditions. In the remainder of this section, we
describe some weaker conditions under which the conclusion of Theorem 3.2.5 still
holds. We will need this more general statement in Section 5 below. For the following
(2)
lemma, we assume only hypotheses 3.1(a) and (b), and we let ∆K be a chain map
from K to K ⊗A K ⊗A K. (We do not assume that there is a coassociative chain map
(2)
∆ : K → K ⊗A K from which ∆K is defined.)
Recall that we have the canonical contracting homotopy GK := πG(ι ⊗A ι) : K ⊗A
K → K for FK (as Lemma 3.2.3 does not require hypothesis 3.1(c)).
Lemma 3.4.1. Assume hypotheses 3.1(a) and (b). For cocycles f, g ∈ HomAe (K, A),
the difference
(2)
ι∗ (π ∗ f ◦ π ∗ g) − f GK (idK ⊗A g ⊗A idK )(π ⊗A π ⊗A π)∆B ι
(2)
(2)
is a boundary. If ∆K = (π ⊗A π ⊗A π)∆B ι, then
ι∗ (π ∗ f ◦ π ∗ g) − f ◦φ g
and
i∗ [π ∗ f, π ∗ g] − [f, g]φ
are boundaries.
Proof. We have
(2)
ι∗ (π ∗ f ◦ π ∗ g) = f πG(id ⊗ gπ ⊗ id)∆B ι
and
(2)
(2)
f GK (idK ⊗ g ⊗ idK )(π ⊗ π ⊗ π)∆B ι = f πG(ι ⊗ ι)(π ⊗ gπ ⊗ π)∆B ι
(2)
= f πG(ιπ ⊗ ιπ)(id ⊗ gπ ⊗ id)∆B ι.
Now one can check that πFB (ιπ ⊗ ιπ) = πFB (since πι = idK and by the definition
of FB ). So
d(πG(ιπ ⊗ ιπ)) = πd(G)(ιπ ⊗ ιπ) = πFB (ιπ ⊗ ιπ) = πFB = d(πG),
LIE BRACKET ON HOCHSCHILD COHOMOLOGY
13
and, since cohomology vanishes in negative degrees, πG(ιπ ⊗ ιπ) − πG is a boundary.
It follows that this difference
(2)
ι∗ (π ∗ f ◦ π ∗ g) − f GK (idK ⊗A g ⊗A idK )(π ⊗A π ⊗A π)∆B ι
(2)
(2)
= f πG(id ⊗ gπ ⊗ id)∆B ι − f πG(ιπ ⊗ ιπ)(id ⊗ gπ ⊗ id)∆B ι
(2)
is a boundary as well, since all of f, π, ι, g, ∆B are cycles.
(2)
(2)
It follows that under hypotheses 3.1(a) and (b), taking ∆K := (π ⊗A π ⊗A π)∆B ι
in Definition 3.2.2, the conclusion of Theorem 3.2.5 holds. Thus φ-brackets may be
defined in a fairly general setting, at the expense of dealing more directly with maps
π, ι comparing to the bar resolution. Note that Definition 3.2.2 can be used to define
(2)
other versions of φ-bracket, given other choices of chain map ∆K . At the moment,
we do not know which of these other φ-brackets are well-defined on cohomology, nor
whether they have useful properties.
As we will see in the example of Section 5, one may be able to produce a satisfactory
(2)
map ∆K without any explicit reference to ι or π, and produce a subsequent candidate
for the Gerstenhaber bracket. In the example of Section 5 we check that our map
(2)
(2)
(2)
∆K is of the form ∆K := (π ⊗A π ⊗A π)∆B ι for some choice of ι and π.
4. Recovering Schouten-Nijenhuis brackets for polynomial rings
4.1. Review of the Koszul resolution. Let A = k[x1 , . . . , xn ] be the polynomial
ring in n variables. We take V to be the k-vector space with basis {x1 , . . . , xn }. As
a formality, let x0 = 1.
Definition 4.1.1. Let Si denote the symmetric group on i symbols. For any v1 , . . . , vi ∈
V , let o(v1 , . . . , vi ) denote the Si -orbit sum
X
sgn(σ)vσ(1) ⊗ . . . ⊗ vσ(i)
o(v1 , . . . , vi ) =
σ∈Si
in V
⊗n
⊂
A⊗n .
We take o(∅) := 1.
Let W denote the graded subspace ⊕i≥0 o(V, . . . , V ) in T A. One can check that W
is a subcoalgebra of T A and that K = K(A) := A ⊗ W ⊗ A is a subcomplex of the
bar resolution B = (A ⊗ T A ⊗ A, d). (See also [4], [16], and (4.2.3) below.) It is well
known that the embedding K → B is a quasi-isomorphism, i.e. that A is Koszul. In
the following lemma, the notation v̂l indicates that vl has been removed.
Lemma 4.1.2. The differential on K is given on monomials by
a ⊗P
o(v1 , . . . , vi ) ⊗ a0
7→ l (−1)l+1 avl ⊗ o(v1 , . . . , v̂l , . . . , vi ) ⊗ a0 − (−1)l+1 a ⊗ o(v1 , . . . , v̂l , . . . , vi ) ⊗ vl a0 .
Proof. This follows by direct computation and the fact that vl vm − vm vl is 0 in A for
each vl , vm ∈ V .
14
CRIS NEGRON AND SARAH WITHERSPOON
We choose the ordering x1 < x2 < · · · < xn on the generators of A and call an
element
1 ⊗ o(xi1 , . . . , xis ) ⊗ xj1 . . . xjt ⊗ o(xk1 , . . . , xku ) ⊗ 1
in k⊗W ⊗A⊗W ⊗k an ordered “monomial” if xil < xil+1 , xjl ≤ xjl+1 , and xkl < xkl+1 ,
for all l. We define ordered monomials in A ⊗ W ⊗ k and in k ⊗ W ⊗ A similarly.
The Ae generating subspaces k ⊗ W ⊗ k and k ⊗ W ⊗ A ⊗ W ⊗ k, of K and K ⊗A K
respectively, are spanned over k by the respective sets of ordered monomials.
We employ a slight variation of a left k-linear contracting homotopy for the identity
on the extended complex K → A → 0 given in [23]. In homological degrees −1 and
0, which are A and A ⊗ A respectively, h is given by the formula
h:
a 7→ 1 ⊗ a
P
xj1 . . . xjt ⊗ a 7→ 1≤ν≤t xj1 . . . xjν−1 ⊗ o(xjν ) ⊗ xjν+1 . . . xjt a,
for any ordered monomial xj1 . . . xjt ⊗1 in A⊗A = K0 and a in A. In higher degrees we
define h to be the right A-linear map specified on ordered monomials by the formulae
7→
(−1)u
P
h : xj1 . . . xjt ⊗ o(xk1 , . . . , xku ) ⊗ 1
xjν >xk xj1 . . . xjν−1 ⊗ o(xk1 , . . . xku , xjν ) ⊗ xjν+1 . . . xjt .
u
When the indexing set {xjν : xjν > xku } is empty, the sum is indeed taken to be 0.
Using Lemma 3.3.1 and the contracting homotopy h given above, one can easily
construct a contracting homotopy φ : K ⊗A K → K for FK in low degrees. One then
deduces from this information the following general formula.
Definition 4.1.3. We define the Ae -linear map φ : K ⊗A K → K on ordered monomials by the formulas
φ : 1 ⊗ xj1 . . . xjt ⊗ o(xk1 , . . . , xku ) ⊗ 1
X
xj1 . . . xjν−1 ⊗ o(xk1 , . . . , xku , xjν ) ⊗ xjν+1 . . . xjt
7→ (−1)u
xku <xjν
φ : 1 ⊗ o(xi1 , . . . , xis ) ⊗ xj1 . . . xjt ⊗ 1
X
7→
xj1 . . . xjν−1 ⊗ o(xjν , xi1 , . . . , xis ) ⊗ xjν+1 . . . xjt
xjν <xi1
on K0 ⊗A Ku and on Ks ⊗A K0 (u, s ≥ 0), and
φ : 1 ⊗ o(xi1 , . . . , xis ) ⊗ xj1 . . . xjt ⊗ o(xk1 , . . . , xku ) ⊗ 1
X
7→
(−1)su+u xj1 . . . xjν−1 ⊗ o(xk1 , . . . xku , xjν , xi1 , . . . , xis ) ⊗ xjν+1 . . . xjt .
xku <xjν <xi1
(4.1.4)
Proposition 4.1.5. The map φ : K ⊗A K → K satisfies d(φ) := dK φ + φdK⊗A K =
FK .
We omit the proof of this proposition, which is a delicate, but straightforward
calculation.
LIE BRACKET ON HOCHSCHILD COHOMOLOGY
15
4.2. Computing the bracket directly from φ and Theorem 3.2.5. Before we
begin let us make a remark. In general, one wants to be strategic in computing
the Lie bracket. One should probably use some additional structures on Hochschild
cohomology, such as the cup product in combination with (2.0.4), additional gradings,
etc. However, we are able to recover here, essentially with a single calculation, a
formula for the brackets of cocycles of arbitrary degree via Theorem 3.2.5.
We will employ the standard isomorphism
A[∂1 , . . . , ∂n ] → HomAe (K, A) = HH(A)
∂i 7→ (a ⊗ o(xj ) ⊗ a0 7→ δij aa0 ).
(4.2.1)
Here the generators ∂i are given degree 1, and A[∂1 , . . . , ∂n ] denotes the free graded
commutative A-algebra on these generators.P We identify the monomial ∂i1 . . . ∂is
s−1
with the function dual to the orbit sum (−1) `=1 ` o(xi1 , . . . , xis ) in HomAe (K, A) =
Homk (W, A). We note that the differential of this complex will vanish, from which
we get the identification HomAe (K, A) = HH(A) of (4.2.1).
It will be convenient to have a bit more notation for the statement of the next
proposition.
Notation 4.2.2. For any ordered set I = {i1 , . . . , is } of integers satisfying 1 ≤ ik ≤ n
for all k, we take
∂I := ∂i1 . . . ∂is ∈ HomAe (K, A)
and
o(xI ) = o(xi1 , . . . , xis ) ∈ W.
For ik ∈ I, we take
I(k) := {i1 , . . . , ik−1 }
and
I 0 (k) := {ik+1 , . . . , il }.
For ordered sets I and J we give I q J the natural ordering with i < j for each
i ∈ I, j ∈ J.
In these notations we do not require that the ordering on I is such that ik < ik+1
as integers. We will always let ik denote the kth element of I, as determined by I’s
given order. If we take n = {1, . . . , n}, then the standard A-basis for A[∂1 , . . . , ∂n ] =
HomAe (K, A) can now be written as the set {∂I : I an ordered subset of n}.
Via indexing by ordered sets, we can give a clear expression of the comultiplication
on W , and the corresponding map ∆ : K → K ⊗A K. We have
X
∆(1 ⊗ o(xI ) ⊗ 1) =
±(1 ⊗ o(xI1 ) ⊗ 1) ⊗ (1 ⊗ o(xI2 ) ⊗ 1),
(4.2.3)
I1 ,I2
where the sum is indexed by all ordered disjoint subsets I1 , I2 ⊂ I with I1 ∪I2 = I, and
± is the sign of σ, where σ is the unique permutation with {iσ(1) , . . . iσ(|I|) } = I1 q I2
as an ordered set.
Proposition 4.2.4. The ◦φ operation is given by
X
∂
(a∂I ) ◦φ (b∂J ) =
(−1)(|I|−k)(|J|−1) a
(b)∂I(k)qJqI 0 (k)
∂xik
1≤k≤|I|
16
CRIS NEGRON AND SARAH WITHERSPOON
and the bracket [ , ]φ is given by
[a∂I , b∂J ]φ =
X
X
∂
∂
(−1)(l−1)(|I|−1) b
(−1)(|I|−k)(|J|−1) a
(b)∂I(k)qJqI 0 (k) −
(a)∂J(l)qIqJ 0 (l) .
∂xik
∂xjl
1≤l≤|J|
1≤k≤|I|
Note that if I and J share some indices, many of the terms a ∂x∂i (b)∂I(k)qJqI 0 (k)
k
may be 0.
Proof. Take f = (a∂I ) and g = (b∂J ) with I and J ordered subsets of n. We may
assume b is an ordered monomial b = xj1 . . . xjt . We first provide a computation with
symbols (−1)i in place of significant signs. We will then go back and provide the
appropriate signs.
Suppose we have a nonzero monomial 1 ⊗ o(xI(k)qJqI 0 (k) ) ⊗ 1. This implies, in
particular, that J and I(k) ∪ I 0 (k) share no indices. Then
∆(2) (1 ⊗ o(xI 0 (k)qJqI(k) ) ⊗ 1) =
(1 P
⊗ o(xI 0 (k) ) ⊗ 1) ⊗ (1 ⊗ o(xJ ) ⊗ 1) ⊗ (1 ⊗ o(xI(k) ) ⊗ 1)
+ PS ±(1 ⊗ o(xI2 ) ⊗ 1) ⊗ (1 ⊗ o(xJ ) ⊗ 1) ⊗ (1 ⊗ o(xI1 ) ⊗ 1)
+ J 0 6=J ±(1 ⊗ o(xI20 ) ⊗ 1) ⊗ (1 ⊗ o(xJ 0 ) ⊗ 1) ⊗ (1 ⊗ o(xI10 ) ⊗ 1),
where S is the set of all pairs of subsets I1 , I2 ⊂ I(k) q I 0 (k) = I \ {ik } with
I1 ∪ I2 = I \ {ik }, minus the pair {I(k), I 0 (k)}. We do not specify the indexing set
of the final sum, except to say that J 0 6= J. Since b∂J (1 ⊗ o(xJ 0 ) ⊗ 1) = 0 whenever
J 0 6= J, the above expression gives
(1 ⊗ (b∂J ) ⊗ 1)∆(2) 1 ⊗ o(xI 0 (k)qJqI(k) ) ⊗ 1 =
X
(−1)1 1 ⊗ o(xI 0 (k) ) ⊗ b ⊗ o(xI(k) ) ⊗ 1 +
±(1 ⊗ o(xI2 ) ⊗ 1) ⊗ b ⊗ (1 ⊗ o(xI1 ) ⊗ 1).
S
Now, one can conclude from the description of S that the maximal element of each
I1 is greater than the minimal element of I2 . So
φ 1 ⊗ o(xI1 ) ⊗ 1) ⊗ b ⊗ (1 ⊗ xI2 ⊗ 1) = 0
and
φ(1 ⊗ (b∂J ) ⊗ 1)∆(2) (1 ⊗ o(xI 0 (k)qJqI(k) ) ⊗ 1)
= (−1)1 φ(1 ⊗ o(xI 0 (k) ) ⊗ b ⊗ o(xI(k) ) ⊗ 1)
= (−1)1 φ(1
P ⊗ o(xI 0 (k) ) ⊗ xj1 . . . xjt ⊗ o(xI(k) ) ⊗ 1)
= (−1)2 xi <xjν <xi
xj1 . . . xjν−1 ⊗ o(xI(k)q{jν }qI 0 (k) ) ⊗ xjν+1 . . . xjt .
k−1
k+1
Finally, since ∂I (1 ⊗ o(xI(k)q{jν }qI 0 (k) ) ⊗ 1) = 0 whenever I(k) q {jν } q I 0 (k) 6= I, i.e.
whenever jν 6= ik , we have
(a∂I ) ◦φ (b∂J ) (1 ⊗ o(xI 0 (k)qJqI(k) ) ⊗ 1)
= a∂I φ(1 ⊗ (b∂J ) ⊗ 1)∆(2) (1 ⊗ o(xI 0 (k)qJqI(k) ) ⊗ 1)
= (−1)3 a |{ν : xjν = xik }| xbi )
= (−1)3 a ∂x∂i (b).
k
k
LIE BRACKET ON HOCHSCHILD COHOMOLOGY
17
One can check that (a∂I ) ◦φ (b∂J ) vanishes on all monomials 1 ⊗ o(xL ) ⊗ 1 with L not
of the form I 0 (k) q J q I(k), after some permutation. So we have
X
∂
(a∂I ) ◦φ (b∂J ) =
(−1)4 a
(b)∂I 0 (k)qJqI(k) ,
∂xik
k
and, after switching the position of I(k) and I 0 (k),
X
∂
(b)∂I(k)qJqI 0 (k) .
(a∂I ) ◦φ (b∂J ) =
(−1)5 a
∂xik
k
As for the signs, we have
P|I|+|J|−2 00
4 = 3 − `00 =1
`
1 = (|I| − k)|J| + `=1 `
P|I|−1
P|J|−1
= 3 − (|I| − 1)(|J| − 1) − `0 =1 `0 − `=1 `
2 = 1 + (|I| − k)(k − 1) + (k − 1)
P|I|−1
5 = 4 + (|I| − 1)|J| + (|I| − k)(k − 1)
3 = 2 + `0 =1 `0
= (|I| − k)(|J| − 1) mod 2.
P|J|−1
The formula for the bracket now follows by the definition of [f, g]φ .
For elements a∂i and b∂j the above formula gives
[a∂i , b∂j ]φ = a
∂
∂
(b)∂j − b
(a)∂i .
∂xi
∂xj
(4.2.5)
One can also verify the identity
[a∂I , b1 ∂J1 b2 ∂J2 ]φ = [a∂I , b1 ∂J1 ]φ b2 ∂J2 + (−1)(|I|−1)|J1 | b1 ∂J1 [a∂I , b2 ∂J2 ]φ .
(4.2.6)
So the bracket given in Proposition 4.2.4 is seen to recover the Schouten-Nijenhuis
bracket, which is generally expressed using the formula (4.2.5) for the bracket on
degree 1 cocycles and extended to all of A[∂1 , . . . , ∂i ] = HH(A) using the graded
derivation identity (4.2.6).
5. Recovering Gerstenhaber brackets for groups of prime order
Assume in this section that the characteristic of the field k is p > 0. Let G be
a cyclic group of order p, with generator g, and A = kG, the group algebra. Let
x := g − 1 in A, so that A ∼
= k[x]/(xp ). The Hochschild cohomology of A is wellknown. See [6] for the algebra structure of HH(kG) in the more general case that G
is abelian. In particular, in that case, HH(kG) ∼
= H(G, k) ⊗ kG as algebras, where
H(G, k) denotes group cohomology. See Sanchez-Flores [19] for the Gerstenhaber
brackets when G is cyclic; using our new techniques, we will recover her results in
our case (i.e. G has order p). While the minimal resolution that we use here does not
satisfy all the hypotheses 3.1(a)–(c) assumed in Theorem 3.2.5, it does satisfy 3.1(a)
and (b) (the weaker conditions assumed in Subsection 3.4). We will show that our
alternative approach yields the Gerstenhaber bracket for these examples.
We will use the following Ae -module resolution of A (see, e.g., [22, Exer. 9.1.4]):
K:
·v
·u
·v
·u
m
· · · −→ Ae −→ Ae −→ Ae −→ Ae −→ A → 0,
where u = x ⊗ 1 − 1 ⊗ x, v = xp−1 ⊗ 1 + xp−2 ⊗ x + · · · + 1 ⊗ xp−1 , and m denotes
multiplication. For each i, let ξi denote the element 1 ⊗ 1 of Ae in degree i.
18
CRIS NEGRON AND SARAH WITHERSPOON
The following maps hn : Kn → Kn+1 constitute a contracting homotopy for K, as
may be verified by direct calculation.
h−1 (xi ) = ξ0 xi ,
i
j
i
j
h0 (x ξ0 x ) =
i−1
X
xl ξ1 xi+j−1−l ,
l=0
h1 (x ξ1 x ) = δi,p−1 xj ξ2 ,
i
j
h2n (x ξ2n x ) = −
j−1
X
xi+j−1−l ξ2n+1 xl
(n ≥ 1),
l=0
h2n+1 (xi ξ2n+1 xj ) = δj,p−1 xi ξ2n+1
(n ≥ 1).
Applying Lemma 3.3.1, we may obtain maps φn : (K ⊗A K)n → Kn+1 , from the
maps hn , for which d(φ) = FK . We only need these maps in degrees 0 and 1:
φ0 (ξ0 ⊗ xi ξ0 ) =
i−1
X
xl ξ1 xi−1−l ,
l=0
i
φ1 (ξ1 ⊗ x ξ0 ) = −δi,p−1 ξ2 ,
φ1 (ξ0 ⊗ xi ξ1 ) = δi,p−1 ξ2 .
Next we record a diagonal map ∆ : K → K ⊗A K. It may be checked directly that
the following map is a chain map:
∆0 (ξ0 ) = ξ0 ⊗ ξ0 ,
∆1 (ξ1 ) = ξ1 ⊗ ξ0 + ξ0 ⊗ ξ1 ,
X
∆2 (ξ2 ) = ξ2 ⊗ ξ0 + ξ0 ⊗ ξ2 +
x a ξ1 ⊗ x b ξ1 x c ,
a+b+c=p−2
∆3 (ξ3 ) = ξ3 ⊗ ξ0 + ξ2 ⊗ ξ1 + ξ1 ⊗ ξ2 + ξ0 ⊗ ξ3 ,
and generally for n ≥ 1,
n−1
n
X
X
X
∆2n (ξ2n ) =
ξ2i ⊗ ξ2n−2i +
xa ξ2i+1 ⊗ xb ξ2n−2i−1 xc ,
i=0 a+b+c=p−2
i=0
∆2n+1 (ξ2n+1 ) =
2n+1
X
ξi ⊗ ξ2n+1−i .
i=0
Let
(2)
∆K := (id ⊗ ∆)∆
(5.0.1)
for the purpose of computing φ-brackets under Definition 3.2.2. (Note that ∆ is
(2)
(2)
coassociative if and only if p = 2.) We will see later that ∆K = (π ⊗ π ⊗ π)∆B ι for
some choice of π and ι, in accordance with Lemma 3.4.1.
We will compute φ-brackets on cohomology in low degrees. Applying HomAe (−, A)
to K, the differentials all are 0. In each degree, the cohomology is the free A-module
A. Let xj ξi∗ ∈ HomAe (Ae , A) denote the function that takes ξi to xj . Cup products
LIE BRACKET ON HOCHSCHILD COHOMOLOGY
19
are known: If p = 2, then ξ1∗ generates the Hochschild cohomology as an A-algebra
(recall HH(A) ∼
= H(G, k)⊗A), while if p > 2, it is generated by ξ1∗ and ξ2∗ . By applying
the identities (2.0.4), we need only compute brackets of pairs of elements of degrees
1 and 2. The φ-circle product of xi ξ1∗ and xj ξ1∗ in degree 1 is given by
(xi ξ1∗ ◦φ xj ξ1∗ )(ξ1 ) = xi ξ1∗ (φ0 (xj ξ0 ⊗ ξ0 + ξ0 ⊗ xj ξ0 + ξ0 ⊗ ξ0 xj ))
= xi ξ1∗ (ξ1 xj−1 + xξ1 xj−2 + · · · + xj−1 ξ1 )
= jxi+j−1 .
Therefore, by symmetry, we obtain
[xi ξ1∗ , xj ξ1∗ ]φ = (j − i)xi+j−1 ξ1∗ .
The φ-circle product of elements in degrees 1 and 2 is given similarly by
(xi ξ1∗ ◦φ xj ξ2∗ )(ξ2 ) = jxi+j−1 ,
while in the reverse order we have
X
(xj ξ2∗ ◦φ xi ξ1∗ )(ξ2 ) = xj ξ2∗
φ1 (−xa ξ1 xb+i ⊗ ξ0 xc ) + φ1 (ξ0 ⊗ xa+b+i ⊗ ξ1 xc )
a+b+c=p−2
=
X
a+c=i−1
= (i + p − i)x
So
degree 2 is
=
jxi+j−1 ξ2∗ .
xi+j−1
a+b=p−1−i
i+j−1
[xi ξ1∗ , xj ξ2∗ ]φ
X
xi+j−1 −
= 0.
Finally, the φ-circle product of two such elements of
(xi ξ2∗ ◦φ xj ξ2∗ )(ξ3 ) = xi ξ2∗ (φ1 (ξ0 ⊗ xj ξ1 ) + φ1 (ξ1 ⊗ xj ξ0 ))
= xi ξ2∗ (δj,p−1 ξ2 − δj,p−1 ξ2 )
= 0.
So [xi ξ2∗ , xj ξ2∗ ]φ = 0. To summarize, in degrees 1 and 2, we have
[xi ξ1∗ , xj ξ1∗ ]φ = (j − i)xi+j−1 ξ1∗ ,
[xi ξ1∗ , xj ξ2∗ ]φ = jxi+j−1 ξ2∗ ,
[xi ξ2∗ , xj ξ2∗ ]φ = 0.
Brackets in higher degrees are determined by these and the identities (2.0.4), since the
Hochschild cohomology is generated as an A-algebra under cup product in degrees 1
and 2.
These computations agree with the Gerstenhaber bracket as computed by SanchezFlores [19], as well as with direct computations of the Gerstenhaber bracket using
standard chain maps ι, π. Next we will verify the conditions of Lemma 3.4.1 to
explain why these φ-brackets agree with Gerstenhaber brackets.
Let ι : K → B and π : B → K be defined as follows. (See [11] for a more general
setting and [5, Section 3] for the maps as below in this specific case.) The chain map
ι is given by
ι2l (ξ2l ) = 1 ⊗ αl and ι2l+1 (ξ2l+1 ) = 1 ⊗ x ⊗ αl
where α0 = 1 and if l ≥ 1,
X
αl =
xi1 ⊗ x ⊗ xi2 ⊗ x ⊗ · · · ⊗ x ⊗ xil+1 .
{i1 +i2 +···+il+1 =lp−l|i1 ,i2 ,...,il ≥1}
20
CRIS NEGRON AND SARAH WITHERSPOON
(Note that in the above sum, il+1 can take on the value 0 while each of i1 , . . . , il must
be at least 1.) The chain map π is given by
π2l (1⊗xi1 ⊗xi2 ⊗· · ·⊗xi2l ⊗1) = ξ2l xi1 +i2 −p xi3 +i4 −p · · · xi2l−1 +i2l −p ,
i1
i2
π2l+1 (1⊗x ⊗x ⊗· · ·⊗x
i2l+1
⊗1) =
iX
1 −1
xm ξ2l+1 xi1 −m−1 xi2 +i3 −p xi4 +i5 −p · · · xi2l +i2l+1 −p .
m=0
(In the above expressions, any term involving a negative exponent of x should be
(2)
interpreted as 0.) Using these maps, we may check directly that the map ∆K defined
(2)
(2)
by (5.0.1) satisfies ∆K = (π ⊗ π ⊗ π)∆B ι. Consequently, Lemma 3.4.1 implies
that φ-brackets, defined as above, coincide with Gerstenhaber brackets, as we have
observed.
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Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803,
USA
E-mail address: cnegron@lsu.edu
Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
E-mail address: sjw@math.tamu.edu